A strichartz inequality for the schrödinger equation on nontrapping asymptotically conic manifolds

We obtain the Strichartz inequality ∫₀¹ ∫_M|u(t, z)|⁴ dt(z)dt ≤ C ||u (0) || _H1/4(M)⁴ for any smooth three-dimensional Riemannian manifold (M, g) which is asymptotically conic at infinity and nontrapping, where u is a solution to the Schrödinger equation iu_t + (1/2)Δ_Mu = 0. The exponent H^1/4(M) is sharp, by scaling considerations. In particular our result covers asymptotically flat nontrapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product U(t, z′, z″) := u(t, z′)u(t, z″) of the solution with itself. We also use smoothing estimates for Schrödinger solutions including one (proved here) with weight r^-1 at infinity and with the gradient term involving only one angular derivative.